The importance of a hybrid coupler as a fundamental passive circuit is demonstrated by its broad employment in many telecommunication systems, both terrestrial and for space applications. Some common examples of the use of such circuits are power splitting networks, distribution networks, duplexers and antenna arrays.
In a typical configuration, a hybrid coupler is formed from several pieces of transmission line with impedances selected to create the desired power splitting and output phase distribution [1]. Very common examples of different types of hybrid coupler are the 90°, 3 dB quadrature coupler and the 180° rat-race coupler. Both of these devices are 2-input, 2-output networks with the property of producing, for the quadrature coupler, a 90° phase shift between the output ports and, for the rat-race coupler, alternatively a 180° or 0° phase shift between the output ports, depending on the chosen input port [1]. In addition, the output power splitting ratio can be arbitrarily adjusted according to the impedance of the transmission lines that form the hybrid coupler impedance [2]-[4].
The quadrature hybrid is generally formed by two coupled quarter-wave transmission lines, 2 straights and 2 shunts. However, more extensive synthesis techniques have been utilized to produce branch-guide couplers that satisfy various desired properties, such as number of branches, power splitting ratio, bandwidth and in-band transfer function [5]-[7].
In recent years, there has been increasing interest regarding the general synthesis of multi-port networks based on coupled resonators [8]-[11]. However, existing fully direct synthesis methods suffer from significant limitations, both in the definition of the polynomials of networks with more than 3 ports, and also for the maximum number of couplings that each resonator can sustain [12].
Modern techniques to synthesize a multi-port circuit, once the rational polynomials for the circuit are known, involve the synthesis of an equivalent transversal network and then the application of a sequence of matrix similarities (matrix rotations) in order to obtain the final topology [10]. This process is based on a conversion from the rational form of the scattering polynomials to the admittance matrix parameters, [Y]ij, expressed as a ratio between the numerators nij and a common denominator, yd, as represented by the following partial fraction expansion notation:
                                          [            Y            ]                    ij                =                                            n              ij                                      y              d                                =                                                    [                                  Y                  ∞                                ]                            ij                        +                                          ∑                                  h                  =                  1                                n                            ⁢                                                r                                      ij                    ,                    h                                                                    s                  -                                      j                    ⁢                                                                                  ⁢                                          λ                      h                                                                                                                              (                  Eq          .                                          ⁢          1                )            where [Y∞]ij is the limit at infinity of the generic element of the admittance matrix, rij,h is the residue associated with pole, λh, the complex low-pass frequency is s=σ+jω, and n is the order of the polynomial of the common denominator yd.
The coupling matrix of a multi-port circuit based on resonators can be defined as
  M  =      [                                        M            p                                                M            pn                                                            M            np                                                M            n                                ]  where Mp is the sub-matrix of the couplings between pairs of external ports, Mpn is the sub-matrix of the coupling coefficients between external ports and internal resonators, and, finally, Mn is the sub-matrix of the coupling coefficients between pairs of internal resonators [13]. From Equation (1) above, the elements of matrices Mp, Mn and Mpn are obtained with direct formulas [13]. The formulas and conversion between the different types of matrices can be performed either analytically for some simple cases [11], or through numerical methods [14]. However, these techniques are valid mainly for multiplexing applications and, in particular, when the transfer function exhibits all single poles, [11]. However, if this last condition is not met, the method based on the derivation of the equivalent transversal network as per Equation (1) above brings singularities to its coupling matrix, thereby leading to a reduction of its columns/rows and thus to the elimination of some ports/resonators (see [11, 12]).